Designing the prediction operator

Figure  shows the form of the desired prediction-error operator in the velocity domain. The prediction lags in the velocity and time directions and the filter size are controlled by the particular characteristics of the dataset, following the same basic designing rules as with the standard 1-D prediction error filter. The coefficients a_ij are defined by a power minimization criterion (Claerbout, 1991). Therefore, if ${\bf S}$ is the semblance of the original data and ${\bf F}$ is the prediction filter, the coefficients of the filter are obtained by minimizing the L₂ error of equation (6), as follows:  

$${\bf F \star S} = 0.$$ (6)

 

predoper Figure 4 The prediction-error filter for suppressing the regions of the velocity semblance spectrum that are associated with primary reflections.

 

realoper Figure 5 The estimated prediction-error filter for the synthetic data of Figure -b.

Once the filter coefficients are defined, the weighting operator ${\bf J_p}$is obtained by  

$${\bf J_p} = {\bf S} - {\bf F \star S}.$$ (7)

Because of the large size of the problem to be solved in equation (6) and since the semblance is a smooth function of time, it is convenient to subsample S in time before solving equation (6) and applying equation (7). The resulting operator ${\bf J_p}$ is then interpolated to the original time sampling interval. The prediction-error filter estimated for the synthetic data of Figure -b is shown in Figure .

Figure  compares the original semblance ${\bf S}$ with the result of applying the prediction operator ${\bf J_p}$ to it for the synthetic dataset of Figure . It is impressive that the filter was able to correctly predict the different multiples patterns present in the data. None of the primaries are visible on the output. Figure -a shows the window function ${\bf W_p}$constructed from ${\bf J_p}$, and Figure -b shows the semblance spectrum of the synthetic data after 15 iterations of the optimization process using ${\bf W_p}$ as the window function. In -b the events associated with reverberations are barely visible. Most importantly, the parts of the spectrum associated with the converted waves were preserved, as show the events at 2.05, 2.5, and 4.3 seconds.

 

sempred
Figure 6 (a) The smoothed semblance spectrum of the data of Figure  and (b) the output of the prediction operator applied to it.

 

semwind
Figure 7 (a)The window function constructed from the predicted semblance spectrum shown in Figure -b. (b) The semblance spectrum of the output of the multiples suppression scheme, using (a) as the window function.

A comparison of the output of the optimization process with the ``ideal" multiple free data (Figure ) shows that not only have the multiples been substantially attenuated, but also the amplitude and phase of the primaries were correctly preserved, including the weak, low-velocity converted waves.

 

predrm
Figure 8 (a) Synthetic data, free of surface related reverberation (same as Figure -a). (b) The output of the multiples suppression scheme applied to the synthetic data of Figure -b, when the prediction-designed function of Figure -a is used as the window operator.